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Existence of solutions with prescribed norm for semilinear elliptic equations. (English) Zbl 0877.35091

The nonlinear eigenvalue problem \[ -\Delta u(x) - g(u(x)) = \lambda u(x),\quad\lambda \in \mathbb{R},\;x \in \mathbb{R}^N \] is studied. It is supposed that \(g:\mathbb{R} \to \mathbb{R}\) is continuous, odd, and such that \[ \alpha \int_0^s g(t) dt \leq g(s)s \leq \beta \int_0^s g(t) dt \] with some \((2N+4)/N<\alpha\leq\beta< (2N)/(N-2)\) if \(N\geq 3\) or \((2N+4)/N<\alpha\leq\beta\) if \(n=1,2\). A certain mild additional assumption is added in the case \(N=1\). It is proved that for any \(c>0\) there exists a weak solution \(u_c \in H^1(\mathbb{R}^N),\;\lambda_c \in \mathbb{R}\) satisfying \(|u|_{L^2}= c,\;\lambda_c <0\). The proof is based on a minimax approach used for the corresponding functional. Further, the dependence of \(|\nabla u_c|_{L^2}\) and \(\lambda_c\) on \(c\) is described. Particularly, it follows \(|\nabla u_c|_{L^2} \to +\infty,\;\lambda_c \to -\infty\) as \(c \to 0\) and \(|\nabla u_c|_{L^2} \to 0,\;\lambda_c \to 0\) as \(c \to +\infty\).
Reviewer: M.Kučera (Praha)

MSC:

35P30 Nonlinear eigenvalue problems and nonlinear spectral theory for PDEs
35J65 Nonlinear boundary value problems for linear elliptic equations
35J20 Variational methods for second-order elliptic equations
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